On the Use of Geometrically Exact Shells for Dynamic Tire Simulation

Abstract

In the present work a tire is modeled by using geometrically exact shells. The discretization is done with the help of isoparametric quadrilateral finite elements. The interpolation is performed with linear Lagrangian polynomials for the midsurface as well as for the director field. As time stepping method for the resulting differential algebraic equation a modified backward differential formula rule is chosen. To handle the interaction with a rigid road surface, a one sided normal contact formulation is introduced. An orthotropic material model for geometrically exact shells derived from 3D continuum theory is introduced, to describe the anisotropic behavior of the tire material. Inflation pressure is taken into account with a configuration dependent force. The interaction between the multibody system of a car and the tire is realized via co-simulation. Some quasi-static simulations are presented and compared to measurements on a real tire.

Appendix: Material Coefficients for the Shell

The isotropic coefficients for the shell material law are given by:

$$\begin{aligned} \mathbb {H}_{\small {\text {iso}}}^{\alpha \alpha \alpha \alpha }&= \frac{E}{1-\nu ^2} G^{\alpha \alpha }G^{\alpha \alpha },\end{aligned}$$

(9.109)

$$\begin{aligned} \mathbb {H}_{\small {\text {iso}}}^{\alpha \alpha \beta \beta }&= \frac{E}{1-\nu ^2} \left( (1-\nu ) G^{\alpha \beta }G^{\alpha \beta } + \nu G^{\alpha \alpha }G^{\beta \beta } \right) ,\end{aligned}$$

(9.110)

$$\begin{aligned} \mathbb {H}_{\small {\text {iso}}}^{\alpha \beta \alpha \beta }&= \frac{E}{1-\nu ^2} \left( (1+\nu ) G^{\alpha \beta }G^{\alpha \beta } + (1- \nu ) G^{\alpha \alpha }G^{\beta \beta } \right) ,\end{aligned}$$

(9.111)

$$\begin{aligned} \mathbb {H}_{\small {\text {iso}}}^{\alpha \alpha \alpha \beta }&=\frac{E}{1-\nu ^2} G^{\alpha \alpha }G^{\alpha \beta },\end{aligned}$$

(9.112)

$$\begin{aligned} \hat{\mathbb {H}}_{\small {\text {iso}}}^{\alpha \beta }&= \frac{E}{2(\nu +1)}G^{\alpha \beta }. \end{aligned}$$

(9.113)

The coefficients of the orthotropic shell material law are given by:

$$\begin{aligned} \mathbb {H}_{\small {\text {orth}}}^{\alpha \alpha \alpha \alpha }&= \sum _{\iota =1}^2 \left( L_1^{\iota \iota } - \frac{L_1^{3\iota }}{L_1^{33}} \right) (B_\iota ^{\alpha })^4 + 2( L_1^{12} + 2L_2^{12}) (B_1^\alpha B_2^\alpha )^2,\end{aligned}$$

(9.114)

$$\begin{aligned} \mathbb {H}_{\small {\text {orth}}}^{\alpha \beta \alpha \beta }&= \sum _{\iota =1}^2 \left( L_1^{\iota \iota } - \frac{L_1^{3\iota }}{L_1^{33}} \right) (B_\iota ^{\alpha }B_\iota ^{\beta })^2 + 4 L_1^{12} B_1^\alpha B_2^\alpha B_1^\beta B_2^\beta \end{aligned}$$

(9.115)

$$\begin{aligned}&\quad + L_2^{12}((B^\alpha _1 B^\beta _2)^2+(B^\alpha _2 B^\beta _1)^2), \end{aligned}$$

(9.116)

$$\begin{aligned} \mathbb {H}_{\small {\text {orth}}}^{\alpha \beta \alpha \beta }&= \sum _{\iota =1}^2 \left( L_1^{\iota \iota } - \frac{L_1^{3\iota }}{L_1^{33}} \right) (B_\iota ^{\alpha }B_\iota ^{\beta })^2 + 2( L_1^{12} + L_2^{12}) (B_1^\alpha B_2^\alpha B_1^\beta B_2^\beta ) \end{aligned}$$

(9.117)

$$\begin{aligned}&\quad + L_2^{12} ((B^\alpha _1 B^\beta _2)^2+(B^\alpha _2 B^\beta _1)^2), \end{aligned}$$

(9.118)

$$\begin{aligned} \mathbb {H}_{\small {\text {orth}}}^{\alpha \alpha \alpha \alpha }&= \sum _{\iota =1}^2 \left( L_1^{\iota \iota } - \frac{L_1^{3\iota }}{L_1^{33}} \right) (B_\iota ^{\alpha })^3B_\iota ^{\beta } \end{aligned}$$

(9.119)

$$\begin{aligned}&\quad + ( L_1^{12} + 2L_2^{12}) ((B_1^{\alpha })^2 B_2^\alpha B_2^\beta + (B_2^{\alpha })^2 B_1^\alpha B_1^\beta ,\end{aligned}$$

(9.120)

$$\begin{aligned} \hat{\mathbb {H}}_{\small {\text {orth}}}^{\alpha \beta }&= \sum _{\iota =1}^2 L_2^{\iota 3}2 B^{\alpha }_{\iota }B^{\beta }_\iota . \end{aligned}$$

(9.121)

Note that we make use of the approximation $G^{\alpha \beta } \approx A^{\alpha \beta }$, such that the integration in the transverse (thickness) direction can be done analytically.